5 A triangle has two congruent angles. The length of one side of the triangle is 16 cm and the length of another
side is 32 cm. Which of the following is the length of
the third side of the triangle?
A. 64 cm
B. 48 cm
C. 32 cm
D. 16 cm
E. It cannot be determined based on the informati

We want to find the length of the missing leg of the given triangle. We will see that the correct options are C and D.

Two angles are congruent if these have the same measure.

Then if the triangle has two congruent angles it means that the triangle must be an isosceles triangle.

We know that one side measures 16cm and the other side measures 32 cm, and this is an isosceles triangle, so it must have two equal sides.

Then the possible measures of the missing side are either 16cm or 32 cm.

So the correct options are C and D.

If you want to learn more about triangles you can read:

link

C. 32 cm

Step-by-step explanation:

Triangle Inequality Theorem

Let y and z be two of the side lengths of a triangle. The length of the third side x cannot be any number. It must satisfy all the following restrictions:

x + y > z

x + z > y

y + z > x

Combining the above inequalities, and provided y>z, the third size must satisfy:

y - z < x < y + z

We know the triangle has two congruent angles, which means the triangle is isosceles, i.e., it has two congruent sides.

We are given two side lengths of 16 cm and 32 cm. The third side must have a length of 16 cm or 32 cm for the triangle to be isosceles.

If the third side had a length of 16 cm then the lengths would be 16-16-32. But that combination cannot form a triangle because of the condition stated above.

If y=16, z=16, and x=32 (the worst possible combination), then the inequality

0 < x < 32

wouldn't be satisfied, thus the third side cannot have a length of 16 cm and it must have a length of 32 cm

C. 32 cm

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